65,078 research outputs found
Error-Correcting Codes for Computer Memories
This thesis is divided into four independent chapters and two appendices.
Chapter I deals with the following generalization of the birthday surprise problem: how many people we need to interview on the average until either r birthdays occur k times each or one birthday occurs k + 1 times. If r = 1, we obtain the usual "birthday surprise" number. We verify that our formula generalizes previous known results. We give asymptotic estimates for the birthday surprise number using a theorem proved in appendix I.
In chapter II, we present accurate and easily evaluated estimates for the average lifetime of a semiconductor RAM memory protected by a single error correcting, doubly error detecting (SEC-DED) code. This problem is somehow related to the one in chapter I. As an application, we give an analysis of the benefits of soft error "scrubbing" when both hard and soft errors are present. We also discuss two methods for increasing the lifetime of a computer memory: adding s rows of spare chips and implementing 2-ECC. We close the chapter by comparing the two methods.
In chapter III, we describe a class of burst error correcting array codes. We prove the fundamental properties of these codes.
Patel and Hong have constructed a code that can correct any track error or two track erasures in a 9-track magnetic tape. In chapter IV, we extend the construction to codes that can correct higher numbers of track errors and erasures. The result is a new family of codes, the B(n,m)-codes.
In appendix I, we prove an important theorem used for asymptotic estimates of integrals. This theorem is used in chapters I and II.</p
Quantum error-correcting codes associated with graphs
We present a construction scheme for quantum error correcting codes. The
basic ingredients are a graph and a finite abelian group, from which the code
can explicitly be obtained. We prove necessary and sufficient conditions for
the graph such that the resulting code corrects a certain number of errors.
This allows a simple verification of the 1-error correcting property of
fivefold codes in any dimension. As new examples we construct a large class of
codes saturating the singleton bound, as well as a tenfold code detecting 3
errors.Comment: 8 pages revtex, 5 figure
Homological Error Correction: Classical and Quantum Codes
We prove several theorems characterizing the existence of homological error
correction codes both classically and quantumly. Not every classical code is
homological, but we find a family of classical homological codes saturating the
Hamming bound. In the quantum case, we show that for non-orientable surfaces it
is impossible to construct homological codes based on qudits of dimension
, while for orientable surfaces with boundaries it is possible to
construct them for arbitrary dimension . We give a method to obtain planar
homological codes based on the construction of quantum codes on compact
surfaces without boundaries. We show how the original Shor's 9-qubit code can
be visualized as a homological quantum code. We study the problem of
constructing quantum codes with optimal encoding rate. In the particular case
of toric codes we construct an optimal family and give an explicit proof of its
optimality. For homological quantum codes on surfaces of arbitrary genus we
also construct a family of codes asymptotically attaining the maximum possible
encoding rate. We provide the tools of homology group theory for graphs
embedded on surfaces in a self-contained manner.Comment: Revtex4 fil
Single-shot fault-tolerant quantum error correction
Conventional quantum error correcting codes require multiple rounds of
measurements to detect errors with enough confidence in fault-tolerant
scenarios. Here I show that for suitable topological codes a single round of
local measurements is enough. This feature is generic and is related to
self-correction and confinement phenomena in the corresponding quantum
Hamiltonian model. 3D gauge color codes exhibit this single-shot feature, which
applies also to initialization and gauge-fixing. Assuming the time for
efficient classical computations negligible, this yields a topological
fault-tolerant quantum computing scheme where all elementary logical operations
can be performed in constant time.Comment: Typos corrected after publication in journal, 26 pages, 4 figure
Analysing correlated noise on the surface code using adaptive decoding algorithms
Laboratory hardware is rapidly progressing towards a state where quantum
error-correcting codes can be realised. As such, we must learn how to deal with
the complex nature of the noise that may occur in real physical systems. Single
qubit Pauli errors are commonly used to study the behaviour of error-correcting
codes, but in general we might expect the environment to introduce correlated
errors to a system. Given some knowledge of structures that errors commonly
take, it may be possible to adapt the error-correction procedure to compensate
for this noise, but performing full state tomography on a physical system to
analyse this structure quickly becomes impossible as the size increases beyond
a few qubits. Here we develop and test new methods to analyse blue a particular
class of spatially correlated errors by making use of parametrised families of
decoding algorithms. We demonstrate our method numerically using a diffusive
noise model. We show that information can be learnt about the parameters of the
noise model, and additionally that the logical error rates can be improved. We
conclude by discussing how our method could be utilised in a practical setting
blue and propose extensions of our work to study more general error models.Comment: 19 pages, 8 figures, comments welcome; v2 - minor typos corrected
some references added; v3 - accepted to Quantu
Check-hybrid GLDPC Codes: Systematic Elimination of Trapping Sets and Guaranteed Error Correction Capability
In this paper, we propose a new approach to construct a class of check-hybrid
generalized low-density parity-check (CH-GLDPC) codes which are free of small
trapping sets. The approach is based on converting some selected check nodes
involving a trapping set into super checks corresponding to a 2-error
correcting component code. Specifically, we follow two main purposes to
construct the check-hybrid codes; first, based on the knowledge of the trapping
sets of the global LDPC code, single parity checks are replaced by super checks
to disable the trapping sets. We show that by converting specified single check
nodes, denoted as critical checks, to super checks in a trapping set, the
parallel bit flipping (PBF) decoder corrects the errors on a trapping set and
hence eliminates the trapping set. The second purpose is to minimize the rate
loss caused by replacing the super checks through finding the minimum number of
such critical checks. We also present an algorithm to find critical checks in a
trapping set of column-weight 3 LDPC code and then provide upper bounds on the
minimum number of such critical checks such that the decoder corrects all error
patterns on elementary trapping sets. Moreover, we provide a fixed set for a
class of constructed check-hybrid codes. The guaranteed error correction
capability of the CH-GLDPC codes is also studied. We show that a CH-GLDPC code
in which each variable node is connected to 2 super checks corresponding to a
2-error correcting component code corrects up to 5 errors. The results are also
extended to column-weight 4 LDPC codes. Finally, we investigate the eliminating
of trapping sets of a column-weight 3 LDPC code using the Gallager B decoding
algorithm and generalize the results obtained for the PBF for the Gallager B
decoding algorithm
Optimal Resources for Topological 2D Stabilizer Codes: Comparative Study
We study the resources needed to construct topological 2D stabilizer codes as
a way to estimate in part their efficiency and this leads us to perform a
comparative study of surface codes and color codes. This study clarifies the
similarities and differences between these two types of stabilizer codes. We
compute the error correcting rate for surface codes and color
codes in several instances. On the torus, typical values are and
, but we find that the optimal values are and . For
planar codes, a typical value is , while we find that the optimal values
are and . In general, a color code encodes twice as much
logical qubits as a surface code does.Comment: revtex, 6 pages, 7 figure
Fault-tolerant logical gates in quantum error-correcting codes
Recently, Bravyi and K\"onig have shown that there is a tradeoff between
fault-tolerantly implementable logical gates and geometric locality of
stabilizer codes. They consider locality-preserving operations which are
implemented by a constant depth geometrically local circuit and are thus
fault-tolerant by construction. In particular, they shown that, for local
stabilizer codes in D spatial dimensions, locality preserving gates are
restricted to a set of unitary gates known as the D-th level of the Clifford
hierarchy. In this paper, we elaborate this idea and provide several extensions
and applications of their characterization in various directions. First, we
present a new no-go theorem for self-correcting quantum memory. Namely, we
prove that a three-dimensional stabilizer Hamiltonian with a
locality-preserving implementation of a non-Clifford gate cannot have a
macroscopic energy barrier. Second, we prove that the code distance of a
D-dimensional local stabilizer code with non-trivial locality-preserving m-th
level Clifford logical gate is upper bounded by . For codes with
non-Clifford gates (m>2), this improves the previous best bound by Bravyi and
Terhal. Third we prove that a qubit loss threshold of codes with non-trivial
transversal m-th level Clifford logical gate is upper bounded by 1/m. As such,
no family of fault-tolerant codes with transversal gates in increasing level of
the Clifford hierarchy may exist. This result applies to arbitrary stabilizer
and subsystem codes, and is not restricted to geometrically-local codes. Fourth
we extend the result of Bravyi and K\"onig to subsystem codes. A technical
difficulty is that, unlike stabilizer codes, the so-called union lemma does not
apply to subsystem codes. This problem is avoided by assuming the presence of
error threshold in a subsystem code, and the same conclusion as Bravyi-K\"onig
is recovered.Comment: 13 pages, 4 figure
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